The snuark algebra is simply a finite subset of the Pauli algebra, the subset generated by spin in the +x, +y, and +z directions. If we were to do our calculations by the usual methods of the Pauli algebra, we would choose an orientation, typically +z, and define the +x and +y states in terms of the +z (spin up) and -z (spin down) states.
Our objective in this post is to relate the snuark algebra to the more traditional ways of calculating with the Pauli algebra. We will do this with the objective of turning the snuark algebra into a QFT, just as qubits can be turned into a QFT. And it will be the QFT theory of snuarks that will allow us to derive the Koide mass formulas.
Our primary application of the snuark algebra is to the particle masses. Even virtual particles have masses, so as far as a QFT goes, to understand mass we need only describe the QFT for the virtual particles. This simplifies things considerably because virtual particles are naturally described in a density matrix form.
In the qubit QFT, the propagators are density matrix states like (1+x)/2. Two oppositely oriented such states add to unity, for example, (1+x)/2 + (1-x)/2 = 1. We can insert this unity into the middle of a propagator without changing the results of the calculation.
For example, let the propagator be (1+z)/2. Then
(1+z)/2 = (1+z)/2 (1+z)/2
= (1+z)/2 [ (1+x)/2 + (1-x)/2 ] (1+z)/2
= (1+z)/2 (1+x)/2 (1+z)/2 + (1+z)/2 (1-x)/2 (1+z)/2
= 0.5(1+z)/2 + 0.5(1+z)/2.
The last line can be obtained in a number of ways. The one most familiar to the audience is to substitute in the Pauli algebra for the density matrices:
Written as Feynman diagrams, putting 1 = (1+x)/2 + (1-x)/2 in the middle of (1+z)/2 amounts to splitting the (1+z)/2 propagator into two halves interupted by the (1+-x)/2 propagators:
These examples are unphysical in that we haven’t given any physical reason why the particle should have its polarity changed. In the physical world, when the orientation of a particle is changed like this, a gauge boson needs to be associated, and a vertex needs to be included in the Feynman diagram. In the above, the vertex was chosen to be 1 to allow the equality to work.
To make a physically realistic reason for a propagator to switch polarity, we need to introduce a gauge boson. Following the example of QED, we will use a vertex of amplitude +- iq, where q is a real number that defines the strength of the interaction. The sign of the interaction will depend on the orientations of the initial and final propagator as follows:
The signs for the process (1+x)/2 to (1+y)/2 is the negative of the sign for the process (1+y)/2 to (1+x)/2 by Hermiticity. The rest of the signs follow by rotating through the even permutations of x, y, and z and so are required by symmetry. Putting this into Feynman diagram form, along with the algebraic notation I prefer and the Pauli notation that is standard in the industry, we have:
These are the basic diagrams that we will use to derive the Koide mass formulas. To reduce products, we will find it very useful to know the following:
The above can be shown by replacing the projection operators with Pauli matrices. We will need it, along with the various results you get by permuting x, y, and z around. The even permutations have the same form, while the odd permutations take a minus sign on the imaginary unit.
The above formula is a specific case of a more general result for three spin projection operators (not necessarily perpendicular as in the above case). The product reduces to the first and last operators, times a complex number whose magnitude is given by the familiar sqrt((1 + cos(theta))/2) formula of particle physics, and whose phase is given by half the surface area of the oriented spherical triangle defined by the three spin projection operators.